# Questions tagged [uniform-spaces]

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### What is the definition of a prorelation?

In the context of quasi-uniform spaces, what is a prorelation? In the text I'm reading, they're defined as a down-directed upper set on relations X->Y. Now, I'm fine with a down-directed up-set, but ...
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### Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
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### Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable. A family $\mathcal F$ of subsets of a topological space ...
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### Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question. Let $X$ be a set with a uniform structure ...
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### Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...
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### Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
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### Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...
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### Cartesian powers of uniform spaces

In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped ...
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### Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. To construct Fine uniformities, Let ...
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### In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...
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### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
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### Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
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### In the category of uniform spaces, is the completion of a quotient map also a quotient map?

I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers. Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous ...
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### The subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) : Let $(X,\mathcal{U})$ be a uniform ...
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### Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
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### Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
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### The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
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### How is the notion of a Lipschitz structure on a manifold defined?

According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is "an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz to ...
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### A theorem of Markov about completely regular spaces and topological groups

In Pontriaguin's classic book Grupos continuos (in English Continuous Groups), says that A. Markov proved that: There are topological groups that are not normal. Furthermore, he says it is deduced ...
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### Extending uniformly continuous functions on subspaces to non-metrizable compactifications

I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$. Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...
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### For any entourage $U,V$ there's an entourage $W$ such that $U\circ W\subseteq V\circ U$

Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true? $$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$ I think if the above ...
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### A uniformity with a countable base is a pseudometric uniformity.

I need a proof for this proposition: If a uniformity $\mathfrak U$ on $X$ has a countable fundamental system of entourages, then it can be defined by a pseudometric on $X$. which is the ...
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### Does every proximal outer measure, measure all open sets?

Let $\: \langle X,\delta\rangle \:$ be a separated proximity space. Let $\: \mu^* \: : \: 2^{X} \: \to \: [0,+\infty] \:$ be a proximal outer measure. Let $U$ be an open subset of $X$. Does it ...
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### Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so? (Original math.SE post) In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (...
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### Does every Lindelof uniform space have a Lindelof completion?

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well known to be true for ...
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### Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
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### Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying: For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural number $n$ ...